We outline a proof that if the Generalized Riemann Hypothesis holds, then every odd number above 5 is a sum of three prime numbers. The proof involves an asymptotic theorem covering all but a finite number of cases, an intermediate lemma, and an extensive computation.

dx.doi.org/10.1090/S1079-6762-97-00031-0
Electronic Research Announcements of the American Mathematical Society
Centrum Wiskunde & Informatica, Amsterdam, The Netherlands

Deshouillers, J.-M, Effinger, G, te Riele, H.J.J, & Zinoviev, D. (1997). A complete Vinogradov 3-primes theorem under the Riemann hypothesis. Electronic Research Announcements of the American Mathematical Society, 3(15), 99–104. doi:10.1090/S1079-6762-97-00031-0